arX

iv:1

103.

0909

v2 [

astr

o-ph

.HE

] 7

Mar

201

1

THE ENVIRONMENT AND DISTRIBUTIONOF EMITTING ELECTRONS

AS A FUNCTION OF SOURCE ACTIVITYIN MARKARIAN 421

Nijil Mankuzhiyil 1

Dipartimento di Fisica, Universita di Udine, via delle Scienze 208, I-33100 Udine (UD), ITALY

Stefano Ansoldi1,2

International Center for Relativistic Astrophysics (ICRA)

Massimo Persic1

INAF-Trieste, via G. B. Tiepolo 11, I-34143 Trieste (TS), ITALY

Fabrizio Tavecchio

INAF-Brera, via E. Bianchi 46, I-23807 Merate (LC), ITALY

Received ; accepted

1and INFN Sezione di Trieste

2and Dipartimento di Matematica e Informatica, Universitadi Udine, via delle Scienze 206,

I-33100 Udine (UD), ITALY

– 2 –

ABSTRACT

For the high-frequency peaked BL Lac object Mrk 421 we study the variation of

the spectral energy distribution (SED) as a function of source activity, from quiescent

to active. We use a fully automatizedχ2-minimization procedure, instead of the “eye-

ball” procedure more commonly used in the literature, to model nine SED datasets

with a one-zone Synchrotron-Self-Compton (SSC) model and examine how the model

parameters vary with source activity. The latter issue can finally be addressed now,

because simultaneous broad-band SEDs (spanning from optical to VHE photon ener-

gies) have finally become available. Our results suggest that in Mrk 421 the magnetic

field (B) decreases with source activity, whereas the electron spectrum’s break energy

(γbr) and the Doppler factor (δ) increase – the other SSC parameters turn out to be un-

correlated with source activity. In the SSC framework theseresults are interpreted in

a picture where the synchrotron power and peak frequency remain constant with vary-

ing source activity, through a combination of decreasing magnetic field and increasing

number density ofγ ≤ γbr electrons: since this leads to an increased electron-photon

scattering efficiency, the resulting Compton power increases, and so does the total (=

synchrotron plus Compton) emission.

Subject headings: BL Lacertae objects: general – BL Lacertae objects: individual (Mrk 421)

– diffuse radiation – gamma rays: galaxies – infrared: diffuse background

– 3 –

1. Introduction

It’s commonly thought that the fueling of supermassive black holes, hosted in the cores

of most galaxies, by surrounding matter produces the spectacular activity observed in AGNs.

In some cases (∼< 10%) powerful collimated jets shoot out in opposite directions at relativistic

speeds. The origin of such jets is one of the fundamental openproblems in astrophysics.

If a relativistic jet is viewed at small angle to its axis, theobserved emission is amplified

by relativistic beaming (Doppler boosting and aberration)allowing deep insight into the

physical conditions and emission processes of relativistic jets. Sources whose boosted jet

emission dominates the observed emission (blazars1) represent a minority among AGN, but

are the dominant extragalactic source class inγ-rays. Since the jet emission overwhelms all

other emission from the source, blazars are key sources for studying the physics of relativistic

extragalactic jets.

The jets’ origin and nature are still unclear. However, it iswidely believed that jets are

low-entropy (kinetic/electromagnetic) flows that dissipate some of their energy in (moving)

regions associated with internal or external shocks. This highly complex physics is approximated,

for the purpose of modelling the observed emission, with oneor more relativistically moving

homogeneous plasma regions (blobs), where radiation is emitted by a non-thermal population

of particles (e.g. Maraschi et al. 1992). The high energy emission, with its extremely fast

and correlated multifrequency variability, indicates that often one single region dominates the

emission.

The jet’s broad-band (from radio toγ-ray frequencies) spectral energy distribution (SED)

is a non thermal continuum featuring two broad humps that peak at IR/X-ray and GeV/TeV

1 Within the blazar class, extreme objects, lacking even signatures of thermal processes usually

associated with emission lines, are defined as BL Lac objects.

– 4 –

frequencies and show correlated luminosity and spectral changes. This emission is commonly

interpreted within a Synchrotron-Self-Compton (SSC) model where the synchrotron and Compton

peaks are produced by one same time-varying population of particles moving in a magnetic field

(e.g., Tavecchio et al. 1998, hencefort T98). The SSC model is perfectly adequate to explain the

SEDs of BL Lac sources (e.g., Tavecchio et al. 2010).

One important issue that should be addressed, but has not yetbeen so far because of the lack

of simultaneous broad-band SEDs, is how the emission changes as a function of the source’s

global level of activity. In particular, given an emission model that fits the data, it is should be

examined what model parameters are correlated with source activity. In order to investigate

SEDs at different levels of activity we choose a high-frequency-peaked BL Lac (HBL) object,

i.e. a blazar (i) whose relativistic jet points directly toward the observer, so owing to relativistic

boosting its SSC emission dominates the source; (ii) whose Compton peak (∼> 100 GeV) can be

detected by Cherenkov telescopes; and (iii) whose GeV spectrum can be described as a simple

power law (unlikely other types of BL Lacs, see Abdo et al. 2009). In addition, such HBL source

must have several simultaneous SED datasets available. Mrk421 meets these requirements. In

this paper we study the variation of its SED with source activity, from quiescent to active.

Another requirement for this kind of study concerns the modeling procedure. We use a

full-fledgedχ2-minimization procedure instead of the ”eyeball” fits more commonly used in the

literature. While the latter at most prove the existence of agood solution, by finely exploring the

parameter space our procedure finds thebest solution and also proves such solution to beunique.

In this paper we investigate the SED of Mrk 421 in nine different source states. To this

aim we fit a one-zone SSC emission model (described in Sect.2), using a fully automatized

χ2-minimization procedure (Sect.3), to the datasets described in Sect.4. The results are presented

and discussed in Sect.5.

– 5 –

2. BL Lac SSC emission

To describe the HBL broad-band emission, we use the one-zoneSSC model of T98. This

has been shown to adequately describe broad-band SEDs of most high-frequency-peaked BL Lac

objects (e.g., Tavecchio et al. 2010) and, for a given source, both its ground and excited states

(Tavecchio et al. 2001; Tagliaferri et al. 2008). The main support for the one-zone model is that

in most such sources the temporal variability is clearly dominated by one characteristic timescale,

which implies one dominant characteristic size of the emitting region (e.g., Anderhub et al. 2009).

Moreover, one of the most convincing evidence favoring the SSC model is the strict correlation

that usually holds between the X-ray and VHEγ-ray variability (e.g., Fossati et al. 2008): since in

the SSC model the emission in the two bands is produced by the same electrons (via synchrotron

and SSC mechanism, respectively), a strict correlation is expected2.

In our work, for simplicity we used a one-zone SSC model, assuming that the entire SED

is produced within a single homogeneous region of the jet. Asalready noted, this class of

models is generally adequate to reproduce HBL SEDs. However, one-zone models also face

some problems in explaining some specific features of TeV blazar emission. In particular,

while very large Doppler factors are often required in one-zone model, radio VLBI observations

hardly detect superluminal motion at parsec scale (e.g., (Piner et al. 2010; Giroletti et al. 2006)).

This led (Georganopoulos & Kazanas 2003; Ghisellini et al. 2005) to propose the existence of

a structured, inhomogeneous and decelerating emitting jet. Inhomogeneous (two-zone) models

(e.g., (Ghisellini & Tavecchio 2008)) have been also invoked to explain the ultra-rapid variability

occasionally observed in TeV blazars (e.g., (Aharonian et al. 2007; Albert et al. 2007).

The emission zone is supposed to be spherical with radiusR, in relativistic motion

2The rarely occurring “orphan” TeV flares, that are not accompanied by variations in the X-ray

band, may arise from small, low-B, high-density plasma blobs (Krawczynski et al. 2004).

– 6 –

with bulk Lorentz factorΓ at an angleθ with respect to the line of sight to the observer, so

that special relativistic effects are cumulatively described by the relativistic Doppler factor,

δ = [Γ(1 − β cos θ)]−1. Relativistic electrons with densityne and a tangled magnetic field with

intensityB homogeneously fill the region.

The relativistic electrons’ spectrum is described by a smoothed broken power-law function of

the electron Lorentz factorγ, with limits γ1 andγ2, break atγbr and low- and high-energy slopes

n1 andn2. This purely phenomenological choice is motivated by the observed shape of the humps

in the SEDs, well represented by two smoothly joined power laws for the electron distribution.

Two ingredients are important in shaping the VHEγ-ray part of the spectrum: (i) using the

Thomson and Klein-Nishina cross sections when, respectively, γhν ≤ mec2 andγhν > mec

2

(with ν the frequency of the ’seed’ photon), in building the model (T98); and (ii) the correction

of & 50GeV data for absorption by the Extragalactic Background Light (EBL), as a function of

photon energy and source distance (e.g., Mankuzhiyil et al.2010, and references therein): for this

purpose we use the popular Franceschini et al. (2008) EBL model.

The one-zone SSC model can be fully constrained by using simultaneous multifrequency

observations (e.g., T98). Of the nine free parameters of themodel, six specify the electron energy

distribution (ne, γ1, γbr, γ2, n1, n2), and 3 describe the global properties of the emitting region

(B, R, δ). Ideally, from observations one can derive six observational quantities that are uniquely

linked to model parameters: the slopes,α1,2, of the synchrotron bump at photon energies below

and above the UV/X-ray peak are uniquely connected ton1,2; the synchrotron and SSC peak

frequencies,νs,C, and luminosities,Ls,C, are linked withB, ne, δ, γbr; finally, the minimum

variability timescaletvar provides an estimate of the source size throughR∼< ctvarδ/(1 + z).

To illustrate how important it is to sample the SED aroundboth peaks, let us consider a

standard situation before Cherenkov telescopes came online, when we would have had only

knowledge of the UV/X-ray (synchrotron) peak. Clearly thiswould have given us information on

– 7 –

the shape of the electron distribution but would have left all other parameters unconstrained: in

particular the degeneracy betweenB andne – inherent in the synchrotron emissivity – could not

be lifted without the additional knowledge of the HE/VHEγ-ray (Compton) peak.

Therefore, only knowledge of observational quantities related to both SED humps enables

determination of all SSC model parameters.

3. χ2 minimization

In this section we discuss the code that we have used to obtainan estimation of the

characteristic parameters of the SSC model. As we recalled in the previous section the SSC model

that we are assuming is characterized by nine free parameters,ne, γ1, γbr, γ2, n1, n2, B, R, δ.

However, in this study we setγ1 = 1, which is a widespread assumption in the literature, so

reducing the number of free parameters to eight.

The determination of the eight free parameters has been performed by finding their best

values and uncertainties from aχ2 minimization in which multi-frequency experimental points

have been fitted to the SSC model described in T98. Minimization has been performed using the

Levenberg–Marquardt method (see Press et al. 1994), which is an efficient standard for non-linear

least-squares minimization that smoothly interpolates between two different minimization

approaches, namely the inverse Hessian method and the steepest descent method.

The algorithm starts by making an educated guess for a starting pointP0 in parameter space

with which the minimization loop is entered; at the same timea (small) constant is defined,

∆χ2

NI(where NI stands for ’Negligible Improvement’), which represents an increment inχ2

small enough that the minimization step can be considered tohave achieved no significative

improvement in moving toward the minimumχ2: this will be used as a first criterion for the exit

condition from the minimization loop. Indeed, to be more confident that the minimization loop

– 8 –

has reached aχ2 value close enough to the absolute minimum, the above condition has to be

satisfied four times in a row: the number of consecutive timesthe condition has been satisfied

is conveniently stored into the integer variablecNI – which is, thus, set to zero at startup. From

the chosen value ofP0 we can compute the associatedχ20= χ2(P0) and enter the minimization

loop. The Levenberg-Marquardt method will determine the next point in parameter space,P ,

whereχ2 will be evaluated. Ifχ2(P ) > χ20

the weight of the steepest-descent method in the

minimization procedure is increased, the variablecNI is set to zero, and we can proceed to the

next minimization step. If, instead,χ2(P ) ≤ χ20

we further check if the decrease inχ2 is smaller

than or equal to∆χ2NI

. If this is the case, the negligible-improvement countercNI is increased

by one: if the resulting value iscNI ≥ 4, we think we have a good enough approximation of the

absolute minimum – and the algorithm ends. If, instead, in the latter testcNI < 4, or if in the

preceding test∆χ2 > ∆χ2NI

, then we increase the weight of the inverse Hessian method inthe

minimization procedure, we setχ20 equal to the lower value we have just found, and we continue

the minimization loop. For completeness and illustration ,we briefly present the flow-chart of the

algorithm in Fig. 1.

A crucial point in our implementation is that from T98 we can only obtain a numerical

approximation to the SSC spectrum, in the form of a sampled SED. On the other hand, at each

step of the loop (see Fig. 1) the calculation ofχ2 requires the evaluation of the SED for all the

observed frequencies. Usually the model function is known analytically, so these evaluations are a

straightforward algebraic process. In our case, instead, we know the model function only through

a numerical sample and it is unlikely that an observed point will be one of the sampled points

coming from the implementation of the T98 model. Nevertheless it will in general fall between

two sampled points, which allows us to use interpolation3 to approximate the value of the SED.

3 The sampling of the SED function derived from T98 is dense enough, so that, with respect to

other uncertainties, the one coming from this interpolation is negligible.

– 9 –

At the same time, the Levenberg–Marquardt method requires the calculation of the partial

derivatives ofχ2 with respect to the eight fitted SSC parameters. Contrary to the usual case, in

which from the knowledge of the model function all derivatives can be obtained analytically,

in our case they have also been obtained numerically by evaluating the incremental ratio of the

χ2 with respect to a sufficiently small, dynamically adjusted increment of each parameter. This

method could have introduced a potential inefficiency in thecomputation, due to the recurrent

need to evaluate the SED at many, slightly different points in parameter space, this being the most

demanding operation in terms of CPU time. For this reason we set up an algorithm to minimize

the number of calls to T98 across different iterations.

4. Datasets

From the literature we then select nine SED datasets corresponding to different emission

states (low to high) of the HBL source Mrk 421.

State 1 and state 2 (Acciari et al. 2009) multi-wavelength campaigns were triggered by

a major outburst in April 2006 that was detected by theWhipple 10 m telescope. A prompt

campaign was not possible because of visibility constraints on XMM-Newton. So simultaneous

multi-wavelength observations took place during the decaying phase of the burst. The optical/UV

and X-ray observations were carried out using XMM-Newton’s optical monitor (OM) and

EPIC-pn detector, respectively. The MAGIC andWhipple telescopes were used for the VHEγ-ray

observations. State 1 strictly simultaneous observationslasted∼4 hrs for state 1, and more than

3 hrs for state 2.

State 3 (Rebillot et al. 2006) reports the multi-wavelengthobservations during December

2002 and January 2003. The campaign was initiated by X-tay and VHE flares detected by the

All Sky Monitor (ASM) of the Rossi X-ray Timing Explorer (RXTE) and the 10 mWhipple

– 10 –

telescope.Whipple and HEGRA-CT1 were used for the VHE observations during the campaign.

Even thoughWhipple observed the source from Dec.4, 2002 to Jan.15, 2003, and HEGRA-CT1

on Nov.3–Dec.12, 2002, only the data taken during nights with simultaneous X-ray observations

are used in this paper to construct the SED. Optical flux is theaverage flux of the data obtained

from Boltwood observatory optical telescope, KVA telescope, and WIYN telescope during the

campaign period.

State 4 and state 9 (Blazejowski et al. 2005) observations were taken from a comparatively

longer time campaign in 2003 and 2004. The X-ray flux obtainedfrom RXTE were grouped

into low-, medium- and high-flux groups. For each X-ray observation in a given group,Whipple

VHE γ-ray data that had been observed within an hour of the X-ray data was selected. State 4 (i.e.,

medium-flux) was observed between March 8 and May 3, 2003; whereas state 9 (i.e., high-flux)

was observed on April 16-20, 2004. Optical datasets obtained with Whipple Observatory’s 1.2 m

telescope and Boltwood Observatory’s 0.4 m telescope were also selected based on the same

grouping method. The optical data measured during the wholecampaign were not simultaneous

with the other multi-wavelength data: however, the opticalflux was not found to vary significantly

during the campaign, so its highest and lowest values are taken to be reliable proxies of the actual

values.

State 5 and state 7 (Fossati et al. 2008) data were taken on March 18-25, 2001 during a

multi-wavelength campaign. State 7 denotes the peak of the March 19 flare, whereas state 5

denotes a post-flare state on March 22 and 23. In both cases theX-ray and VHEγ-ray data were

obtained with, respectively,RXTE and theWhipple telescope. The lowest and highest optical

fluxes obtained during the whole campaign with the 1.2 m Harvard-Smithsonian telescope on

Mt. Hopkins were used in the SEDs for states 5 and 7, respectively.

State 6 (Acciari et al. 2009) observations were taken, usingthe same optical and X-ray

instruments as states 1 and 2, during a decaying phase of an outburst in May 2008. The VHEγ-ray

– 11 –

data were taken with VERITAS. There are∼2.5 hours of strictly simultaneous data.

State 8 (Donnarumma et al. 2009) data were taken during a multi-wavelength campaign on

June 6, 2008: VERITAS,RXTE and Swift/BAT, and WEBT provided the VHEγ-ray, X-ray, and

optical data, respectively.

5. Results and discussion

To each of the datasets we apply ourχ2-minimization procedure (see Fig. 1). The best-fit

SSC models are plotted alongside the SED data in Fig. 2.

Obtaining truly best-fit SSC models of simultaneous blazar SEDs is crucial to measure

the SSC parameters describing the emitting region. The obtained optimal model is proven to

be unique because the SSC manifold is thoroughly searched for the absoluteχ2 minimum.

Furthermore, the very nature of our procedure ensures that there is no obvious bias affecting the

resulting best-fit SSC parameters.

Nevertheless, as it has also been discussed (Andrae et al. 2010), there may be caveats

related with theχ2ν fitting, especially when applied to non-linear models such as the present one.

For this reason, it is important to try to understand the goodness of the fit with methods other

than the value ofχ2ν (reported in table 1). Following (Andrae et al. 2010), we have applied the

Kolmogorov-Smirnov (KS) test for normality of the residuals of all our SED fits. A standard

application of the test shows that in all cases the residualsarenot normally distributed: the KS

test thus fails at the 5% significance level. It is, of course,crucial to understand the reason for

this behavior of our SSC fits with respect to the KS test. Let usstart with some general remarks

about the modelling of blazar emission and their observations. First, the one zone SSC model

contains two distinct physical processes in one same region, i.e. synchrotron emission and its

Compton up-scattered counterpart, that manifest themselves as essentially separate components

– 12 –

at very different energies; on the other hand, additional subtle effects may enter the modelling of

blazar emission, so that the SSC model may only be an approximation to the real thing. (A more

refined KS analysis suggests exactly this, see below). Second, our blazar datasets do cover the (far

apart) energy ranges spanned by, respectively, the synchrotron and Compton emission: but they

markedly differ in these two spectral regions, in that the uncertainties associated with VHE data

are much larger than those associated with the optical and X-ray data.

Both these observations suggest us and give us the possibility of a slightly different approach

to the problem of the statistical significance of the fits, which will turn out to be quite enlightening.

We will refer to this other approach as thepiecewise KS test: it consists in applying the KS test,

separately to low energy and high energy data (see Appendix). The main motivation behind this

idea is, as discussed above, the marked difference, from both the physical and the data-quality

point of view, of the lower and higher energy ranges. Surprisingly, if for each SED we separately

check the low- and high-energy residuals for normality, theKS test always confirms their

(separate) normality at the 5% confidence level. On the technical side we took all the necessary

precautions to improve reliability: the null hypothesis statistics have been obtained from a

Montecarlo simulation of 100,000 datasets having the same dimension as the residuals datasets.

Critical values to test normality of the residuals have beenobtained from these numerically

determined statistics and, normality holds in all that cases we have considered. Hence, it is

unlikely to be a coincidence, and this calls for an explanation. Clearly, the fact that the piecewise

KS test is not able to reject the normality of the low-/high-energy residuals separately, means that

the fitted SEDs can be considered as a reasonable models separately at low and high energies. Of

course, because of the quality of especially high energy data, the uncertainties on the parameters

of the fit are sometimes quite large, a fact that unfortunately can not be avoided, despite the fact

that the quality of the datasets that we are using is certainly above average among those that are

available. At the same time, the failure for normality of theresiduals on the standard KS test, in

which low- and high-energy residuals are considered simultaneously, suggests that existing data

– 13 –

may require our adopted SSC model to be improved – perhaps by taking into account higher-order

effects (e.g., a multiple-break or curved electron spectrum). So, and with more and better VHE

data available, it could be possible to reduce the uncertainties in the parameters and to obtain a

higher overall significance in the fit.

Given the above, the results found with our fits are the best possible in the framework of

existing datasets and models of blazar emission: at least, although preliminary, they have a

quantitative and clear statistical meaning. Therefore, wethink it is useful to examine some of their

possible consequences. We’ll do so in what follows.

In Tables 1, 2 we report the best-fit SSC parameters. Source activity (measured as the total

luminosity of the best-fit SSC model) appears to be correlated with B, γbr, andδ (see Fig. 3-top)

– and to be uncorrelated with the remaining SSC parameters. The bolometric luminosity used

in these plots has been obtained directly from the fitted SED.In more detail, after determining

the numerical approximation to the SEDlog[νF (ν)], the parameters being fixed at their best

values obtained with the previously described minimization procedure, we have performed

L =∫ νmax

νmin

νF (ν)dν, with νmin, νmax set at2.5 decades, respectively, below the synchrotron peak

and above the Compton peak. In this way we make sure to performthe integral over all the

relevant frequencies in a way that is independent from any inlocation of these peaks.

We then searched the data plotted in the top row of Fig. 2 for possible correlations. The

linear-correlation coefficients turn out to be0.67, 0.64 and0.54, which confirm linear correlations

with confidence levels of4.8%, 6.3% and13.3%, respectively. As an additional test (given the

relatively low statistics of our datasets), we checked thatthe KS test confirms normality of the fit

residuals4.

4The same approach involving a Montecarlo generated empirical distribution for the null-

hypothesis described just above has been used also in all these cases.

– 14 –

All the parameters derived through our automatic fitting procedure are within the range

of SSC parameters found in the literature for HBLs in general(e.g. (Tavecchio et al. 2001,

2010; Tagliaferri et al. 2008; Celotti & Ghisellini 2008)) and for Mrk 421 in particular (e.g.,

(Bednarek & Protheroe 1997; Tavecchio et al. 1998; Maraschiet al. 1999; Ghisellini et al. 2002;

Konopelko et al. 2003; Fossati et al. 2008)). In particular,large Doppler factors such as those

derived in our extreme cases,δ > 50, have been occasionally derived (e.g., (Konopelko et al.

2003; Fossati et al. 2008); see also the discussion in (Ghisellini et al. 2005)).

As is seen from Fig. 3-top, γbr andB are correlated, respectively, directly (left) and inversely

(right) with L. This may be explained as follows. An increase ofγbr implies an effective increase

of the energy of most electrons (or, equivalently, of the density of γ < γbr electrons). To keep the

synchrotron power and peak roughly constant (within a factor of 3; see Fig. 2),B must decrease.

This improves the photon-electron scattering efficiency, and the Compton power increases. The

total (i.e., synchrotron plus Compton) luminosity will be higher. So a higherγbr implies a lower

B and a higher emission state. Theδ–L correlation (Fig. 3–top-middle) results from combining

theB–δ inverse correlation (Fig. 3–bottom-right; see below) and theB–L anticorrelation.

A deeper insight on emission physics can be reached plottingthe threeL-dependent

parameters one versus the other (Fig. 3-bottom). TheB–γbr anticorrelation, with∆logB ≃

−2∆log γbr (Fig. 3–bottom-left), derives from the synchrotron peak,νs ∝ Bγ2, staying roughly

constant (see Fig. 2). For fixed synchrotron and Compton peakfrequencies in a relativistically

beamed emission, theB–δ relation is predicted to be inverse in the Thompson limit anddirect

in the Klein-Nishina limit (e.g., Tavecchio et al. 1998): becauseνs, νc do not greatly vary

from state to state in our data (see Fig. 2), the correlation in Fig. 3–bottom-right suggests that

the Compton emission of Mrk 421 is always in the Thompson limit. Theδ–γbr correlation

(Fig. 3–bottom-middle) results as a corollary of the condition of constantνs, νc emitted by a

plasma in bulk relativistic motion toward the observer.

– 15 –

Our fits show clear trends among some of the basic physical quantities of the emitting region,

the magnetic field, the electron Lorentz factor at the spectral break, and the Doppler factor (see

Fig. 3). In particular,B andγbreak follow a relationB ∝ γ−2

break, whileB andδ are approximately

related byB ∝ δ−2.

Rather interestingly, the relation connectingB andγ is naturally expected within the context

of the simplest electron acceleration scenarios (e.g., (Henri et al. 1999)). In this framework, the

typical acceleration timescale,tacc, is proportional to the gyroradius:tacc(γ) ∝ rL/c, where

rL = γmec/(eB) is the Larmor radius. On the other hand, acceleration competes with radiative

(synchrotron and IC) cooling. In Mrk 421, characterized by comparable power in the synchrotron

and IC components, we can assume thattcool(γ) ≈ tsyn ∝ 1/γB. The maximum energy reached

by the electrons is determined by setting these two timescales equal, i.e.

tacc(γmax) = tcool(γmax) → γmax ∝ B−1/2 , (1)

in agreement with the relation derived by our fit. It is therefore tempting to associateγbreak to γmax

and explain theB ∝ γ−2

breakrelation as resulting from the acceleration/cooling competition.

If the above inference is correct, one can explain the variations ofγbreak as simply reflecting

the variations ofB in the acceleration region. In turn, the variations ofB could be associated

either to changes in the global quantities related to the jetflow or to some local process in the jet

(i.e. dissipation of magnetic energy through reconnection). The second relation mentioned above,

i.e. δ vs.B, seems to point to the former possibility. First of all let usassumeδ ∼ Γ (since we are

probably observing the Mrk 421 jet at a small angle w.r.t. theline of sight). A general result of jet

acceleration models is that, during the acceleration phase, the jet has a parabolic shape,R ∝ d1/2

(with d the distance from the central black hole), and the bulk Lorentz factorΓ increases withd

asΓ ∝ d1/2 (e.g., (Vlahakis & Konigl 2004; Komissarov et al. 2007). Onthe other hand, if the

magnetic flux is conserved,B ∝ R−2 ∝ d−1 and thus we foundB ∝ Γ−2. Albeit somewhat

speculative, these arguments suggest that the trends displayed in Fig. 3 are naturally expected in

– 16 –

the general framework of jet acceleration.

The correlations in Fig. 3-bottom seem to be tighter than those in Fig. 3-top. The larger

scatter affecting the latter owes to the fact that the electron density,ne, that also enters the

definition of SSC luminosity, shows no correlation withB, γbr, andδ – hence it slightly blurs the

latter’s plots with luminosity.

One further note concerns error bars. Our code returns1-σ error bars. To our best knowledge,

this is the first time that formal errors of SED fits are obtained in a rigorous way. As an example

of the soundness of the method, note that the obtained valuesof δ are affected by the largest errors

when the distribution of the VHE data points is most irregular (e.g. states 4, 7).

We notice that the variability of Mrk 421 markedly differs from that of (e.g.) the other nearby

HBL source, Mrk 501. The extremely bursting state of 1997 showed a shift ofνs andνc by two

and one orders of magnitude, respectively, suggesting a Klein-Nishina regime for the Compton

peak (Pian et al. 1998). Based on the data analyzed in this paper, Mrk 421 displays (within

our observational memory) a completely different variability pattern. However, one important

similarity may hold between the two sources: based on eyeball-fit analysis of Mrk 501’s SED in

different emission states using a SSC model similar to the one used here, Acciari et al. (2010)

suggest thatγbr does vary withL. If this correlation is generally true in blazars, the implication

is that particle acceleration, providing fresh high-energy electrons within the blob, must be one

defining characteristic of excited source states.

We thank Daniel Gall for providing the datasets corresponding to states 1, 2 and 6, and an

anonymous referee for useful comments and suggestions. Oneof us (SA) acknowledges partial

support from the long-term Workshop on Gravity and Cosmology (GC2010: YITP-T-10-01) at

the Yukawa Institute, Kyoto University, during the early stages of this work, and warmly thanks

P. Creminelli and S. Sonego for insightful discussions on some topics touched upon in this paper.

– 17 –

START

assign P0, initial pointin SSC parameter space,and ∆χ2

NI; set cNI = 0

compute χ2(P0)χ2

0= χ2(P0)

determine P , next pointin SSC parameter space

compute χ2(P )

increase weight ofsteepest descent

χ20

= χ2(P )increase weight ofinverse Hessian

cNI = 0χ2

0= χ2(P )

χ2(P )− χ20

< ∆χ2

NI

cNI = cNI + 1cNI < 4

output values of χ2,

SSC parameters andassociated uncertainties

END

χ2(P ) ≥ χ2

0

χ2(P ) < χ2

0

NO

YESYES

NO

Fig. 1.— Flow chart of the minimization code.

Source B R δ χ2ν

[gauss] [cm]

State 1 (9± 3)× 10−1 (9± 4)× 1014 (2 ± 0.5)× 101 0.84

State 2 (8± 6)× 10−1 (8± 4)× 1014 (2.7± 1.1)× 101 1.86

State 3 (6± 6)× 10−2 (2.0± 1.5)× 1015 (1.0± 0.5)× 102 0.91

State 4 (1.21± 0.16)× 10−1 (1.1± 1.3)× 1015 (8± 6)× 101 0.89

State 5 (1.9± 1.3)× 10−1 (10± 4)× 1014 (7± 5)× 101 0.67

State 6 1.0± 0.7 (6± 3)× 1014 (2.8± 1.1)× 101 1.39

State 7 (4± 3)× 10−2 (2± 5)× 1015 (8± 7)× 101 1.61

State 8 (6± 3)× 10−2 (2± 1.8)× 1015 (1.1± 0.4)× 102 0.60

State 9 (4± 3)× 10−2 (2± 4)× 1015 (1.2± 1.0)× 102 0.85

Table 1: Best-fit single-zone SSC model parameters for the nine datasets of Mrk 421. States are

named as in Fig. 2.

– 18 –

-14

-12

-10

-8

April 2006

state 1

April 2006

state 2

December 2002

state 3

-14

-12

-10

-8

January-March 2004

state 4

March 2001

state 5

May 2008

state 6

15 20 25

-14

-12

-10

-8

March 2001

state 7

15 20 25

June 2008

state 8

15 20 25

April 2004

state 9

Fig. 2.— Best-fit one-zone SSC models for nine data sets referring to different emission lev-

els of the HBL source Mrk 421. Source states are ordered by increasing model luminosity and

data have been obtained as follows: state 1 (Acciari et al. 2009), state 2 (Acciari et al. 2009),

state 3 (Rebillot et al. 2006), state 4 (Blazejowski et al. 2005), state 5 (Fossati et al. 2008), state

6 (Acciari et al. 2009), state 7 (Fossati et al. 2008), state 8(Donnarumma et al. 2009), state 9

(Blazejowski et al. 2005).

– 19 –

Fig. 3.—Top. Variations of the SSC parameters magnetic fieldB, Lorentz factorδ, andγbr as a

function of the model’s bolometric luminosity. The other SSC parameters that are left to vary (R,

ne, γ2, n1, n2 only show a scatter plot with the luminosity.Bottom. Correlations betweenB, δ, and

γbr.

– 20 –

APPENDIX

A TOY MODEL MIMICKING THE PIECEWISE KS TEST RESULTS FOR THE SED’S

In this appendix we reproduce the outcome of the piecewise KSapproach to goodness of fit

using a simulated toy model. Let us consider the functionp(x; a, b, c) := ax4 + bx2 + c. By using

a pseudo-random generator, that produces uniformly distributed pseudo-random numbers in the

interval[0, 1] – which we will henceforth callrnd –, we generated(xi, yi)i=1,...,5 pairs of points and

(wi)i=1,...,5 numbers, wherexi = −2.3+0.1·(i+0.1·rnd), yi = p(−2.3+0.1·i;−1, 4, 0)+0.2·rnd

andwi = σ−2

i with σi = 0.01 · (rnd+ 1). These can be considered as5 measurements around the

x = −2 local maximum of the function−x4 + 4x2 with small uncertaintiesσi. We then generated

another5 pairs of points(xi, yi)i=6,...,10 and(wi)i=6,...,10 numbers withxi = 1.7+0.1·(i+0.1·rnd),

yi = p(1.8 + 0.1 · i;−2, 16,−14) + 0.3 · rnd, wi = σ−2

i and, finally,σi = 0.1 · (rnd+ 1), which

again can be considered as5 slightly randomized measurement around a maximum, but now the

maximum of the function−2x4 + 8x2 + 1 and with much larger uncertainties that the first five

ones.

By minimizingχ2 we can fit the above set of10 randomly generated data points versus the

functionp(x; a, b, c). After obtaining the desired fitted parameters we calculatethe residuals and,

using the KS test, check if they are normally distributed. The KS test rejects normal distribution

of the entire set of residuals at the 5% significance level. Onthe other hand, if we separately apply

the KS test to the first, i.e.{i = 1, . . . , 5}, and the second,{i = 6, . . . , 10}, subset of residuals,

the test confirms that the residuals are normally distributed. This exactly replicates the behavior

we obtained in the SED fits.

– 21 –

Source ne γbr γmax n1 n2

[cm−3]

State 1 (1.3± 1.5)× 103 (2.6± 0.9)× 104 (1.05± 0.18)× 107 1.49± 0.19 3.77± 0.11

State 2 (1 ± 3)× 103 (2.4± 0.9)× 104 (4.1± 1.1)× 106 1.5± 0.3 3.62± 0.14

State 3 (5 ± 5)× 103 (7± 3)× 104 (7± 5)× 107 2.05± 0.10 4.8± 0.3

State 4 (2 ± 5)× 103 (4± 2)× 104 (8.2± 1.7)× 106 1.8± 0.3 4.11± 0.13

State 5 (2 ± 5)× 103 (4.5± 1.9)× 104 (2.4± 0.3)× 107 1.7± 0.3 4.3± 0.180

State 6 (4 ± 4)× 103 (1.9± 0.6)× 104 (1.8± 0.4)× 106 1.54± 0.11 4.37± 0.09

State 7 (1 ± 7)× 103 (8± 6)× 104 (7± 2)× 106 1.7± 0.4 4.23± 0.20

State 8 (4 ± 9)× 101 (5± 2)× 104 (1.6± 0.4)× 107 1.5± 0.2 4.22± 0.14

State 9 (1 ± 7)× 102 (8± 9)× 104 (1.1± 0.4)× 107 1.6± 0.5 3.9± 0.2

Table 2: Best-fit single-zone SSC model parameters for the nine datasets of Mrk 421. Numbering

convention as in Fig. 2.

– 22 –

REFERENCES

Abdo, A.A., et al. (Fermi/LAT Collaboration) 2009, ApJ, 707, 1310

Acciari, V. A., et al. (VERITAS Collaboration) 2009, ApJ, 703, 169

Acciari, V.A. et al. (VERITAS and MAGIC Collaborations) 2010, ApJ, submitted

Aharonian, F., et al. 2007, ApJL, 664, L71

Albert, J., et al. (MAGIC Collaboration) 2007, ApJ, 669, 862

Anderhub, H., et al. (MAGIC collaboration) 2009, ApJ, 705, 1624

Andrae, R., Schulze-Hartung, T., & Melchior, P., 2010, arXiv:1012.3754v1 [astro-ph.IM]

Bednarek, W., & Protheroe, R.J. 1997, MNRAS, 292, 646

Blazejowski M. et al. 2005, ApJ, 630, 130

Celotti, A., & Ghisellini, G. 2008, MNRAS, 385, 283

Donnarumma I. et al. 2009, ApJ, 691, L13

Fossati, G., Buckley, J.H., Bond, I.H., et al. 2008, ApJ, 677, 906

Franceschini, A., Rodighiero, G., & Vaccari, M. 2008, A&A, 487, 837

Georganopoulos, M., & Kazanas, D. 2003, ApJL, 594, L27

Ghisellini, G., Celotti, A., & Costamante, L. 2002, A&A, 386, 833

Ghisellini, G., Tavecchio, F., & Chiaberge, M. 2005, A&A, 432, 401

Ghisellini, G., & Tavecchio, F. 2008, MNRAS, 386, L28

Giroletti, M., Giovannini, G., Taylor, G.B., & Falomo, R. 2006, ApJ, 646, 801

– 23 –

Henri, G., Pelletier, G., Petrucci, P.O., & Renaud, N. 1999,Astrop. Phys., 11, 347

Komissarov, S.S., Barkov, M.V., Vlahakis, N., & Konigl, A.2007, MNRAS, 380, 51

Konopelko, A., Mastichiadis, A., Kirk, J., de Jager, O.C., &Stecker, F.W. 2003, ApJ, 597, 851

Krawczynski, H., Hughes, S.B., Horan, D., et al. 2004, ApJ, 601, 151

Mankuzhiyil, N., Persic, M., & Tavecchio, F. 2010, ApJL, 715, L16

Maraschi, L., et al. 1999, ApJL, 526, L81

Maraschi, L., Ghisellini, G., & Celotti, A. 1992, ApJ, 397, L5

Pian, E., et al. 1998, ApJ, 492, L17

Piner, B.G., Pant, N., & Edwards, P.G. 2010, ApJ, 723, 1150

Press, W.H., et al. 1992, Numerical Recipes (Cambridge: Cambridge University Press)

Rebillot, P. et al. 2006, ApJ, 641, 740

Tagliaferri, G., et al. (MAGIC collab.) 2008, ApJ, 679, 1029

Tavecchio, F., et al. 2001, ApJ, 554, 725

Tavecchio, F., Maraschi, L., & Ghisellini, G. 1998, ApJ, 509, 608 (T98)

Tavecchio, F., et al. 2010, MNRAS, 401, 1570

Vlahakis, N., & Konigl, A. 2004, ApJ, 605, 656

This manuscript was prepared with the AAS LATEX macros v5.2.

-14

-12

-10

-8

step 1 step 2 step 3

-14

-12

-10

-8

step 4 step 5 step 6

15 20 25

-14

-12

-10

-8

step 7

15 20 25

step 8

15 20 25

step 9